# Langevin equation

(Difference between revisions)
 Revision as of 17:29, 2 November 2005 (view source)Salva (Talk | contribs)← Older edit Revision as of 17:30, 2 November 2005 (view source)Salva (Talk | contribs) mNewer edit → Line 6: Line 6: where $dW(t)$ is a Wiener process. where $dW(t)$ is a Wiener process. - $u'$ is the turbulence intensity and math>  \tau [/itex] ia Lagrangian time-scale. + $u'$ is the turbulence intensity and   \tau [/itex] a Lagrangian time-scale. Th finite difference  approximation of the above equation is Th finite difference  approximation of the above equation is

## Revision as of 17:30, 2 November 2005

The stochastic differential equation (SDE) for velocity component $U(t)$, the Langevin equation is

$dU(t) = - U(t) \frac{dt}{\tau} + \frac{2 u'}{\tau}^{1/2} dW(t)$

where $dW(t)$ is a Wiener process. $u'$ is the turbulence intensity and $\tau$ a Lagrangian time-scale.

Th finite difference approximation of the above equation is

$U(t+\Delta t) = U(t) - U(t) \frac{\Delta t}{\tau} + \frac{2 u' \Delta t}{\tau}^{1/2} \mathcal{N}$

where $\mathcal{N}$ is a standardized Gaussian random variable with 0 mean an unity variance which is independent of $U$ on all other time steps (Pope 1994). The Wiener process can be understood as Gaussian random variable with 0 mean and variance $dt$